All of the 3rd grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$2.50$ each for students, and the group paid $$43.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$18.00$ each for teachers and $$12.50$ each for students, and the group paid $$179.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+2.5y = 43}$ ${18x+12.5y = 179}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-18x-7.5y = -129}$ ${18x+12.5y = 179}$ Add the top and bottom equations together. $ 5y = 50 $ $ y = \dfrac{50}{5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6x+2.5y = 43}$ to find $x$ ${6x + 2.5}{(10)}{= 43}$ $6x+25 = 43$ $6x = 18$ $x = \dfrac{18}{6}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {18x+12.5y = 179}$ and get the same answer for $x$ ${18x + 12.5}{(10)}{= 179}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.